A critical point theorem for a class of non-differentiable functionals with applications

1.   Introduction

The mountain pass theorem of Ambrosetti and Rabinowitz for $C^{1}$ functions plays an essential role in the area of nonlinear analysis. We are interested in the monograph ^[22] of Motreanu and Panagiotopoulos in which they established a new version of the mounbtain pass theorem [22,Theorem 3.2] for the functionals $f$ from Banach space $X$ to ${\mathbb{R}}\cup \{+\infty\}$ satisfying the following hypothesis:

$(H_f): f(x) = \Phi(x)+\Psi(x)$ for all $x\in X$, where $\Phi:X\rightarrow {\mathbb{R}}$ is locally Lipschitz continuous while $\Psi:X\rightarrow {\mathbb{R}}\cup \{+\infty\}$ is convex, proper, and lower semi-continuous.

We call $x\in X$ is a critical point of $f$ if $x$ solves the following problem:

where $\Phi^{0}(x; z-x)$ is the generalized directional derivative of $\Phi$ at $x$ in the direction $z-x$ (see ^[6] for detail).

Recall that $f$ satisfies the $(PS)_c$ condition if any sequence $\{x_{n}\}\subset X$ for which $\lim\limits _{n\rightarrow + \infty} f(x_{n}) = c$ and

where $\epsilon_{n}\rightarrow 0^{+}$, possesses a convergent subsequence.

When $(PS)_c$ holds true at any level $c$ we simply write $(PS)_f$ in place of $(PS)_c$.

Inequality (1.1) is usually called variational-hemivariational inequality, which has been exploited for mathematically formulating several engineering, besides mechanical questions and extensively studied from many points of view in the latest years ^[1,22,23].

Variational-hemivariational inequalities can be studied in the framework of a general critical point theory which combines features of the classical convex analysis and of the theory of generalized gradients for locally Lipschitz functions. Such inequalities represent a very general pattern for several kinds of variational problems. Indeed, if $\Phi\in C^{1}(X, {\mathbb{R}})$, the problem (1.1) is reduced to a variational inequality and the relevant critical point theory as well as significant applications are developed in ^[25]; if $\Psi\equiv0$, then (1.1) coincides with the problem treated by Chang in ^[5] which is called differential equations with discontinuous nonlinearities; differential inclusions (see ^[12]) and special non-smooth problems with constraints (see ^[13]) can be considered as special cases of variational-hemivariational inequality. Finally, when both $\Phi\in C^{1}(X, {\mathbb{R}})$ and $\Psi\equiv0$, the problem (1.1) becomes the Euler equation $\Phi'(u) = 0$ and the theory is classical. For the new results on this topic, see the excellent overview in ^{[4,7,10,11,14,15,17,18,19,26,27]}.

Chang in ^[5] established the critical point theory for non-differentiable functionals and represented some applications to partial differential equations with discontinuous nonlinearities. Marano and Motreanu ^[20] obtained a critical points theorem which extends the variational principle of Ricceri to variational-hemivariational inequalities and semilinear elliptic eigenvalue problems with discontinuous nonlinearities. The critical point theorem in presence of splitting was established by Br$\grave{e}$zis-Nirenberg ^[3]. Subsequently Livrea, Marano and Motreanu ^[16] extended it to Motreanu-Panagiotopoulos' setting under the the following structural hypothesis $(H_f)'$:

$(H_f)'$: $f(x) = \Phi(x)+\Psi(x)$ for all $x\in X$, where $\Phi:X\rightarrow {\mathbb{R}}$ is locally Lipschitz continuous while $\Psi:X\rightarrow {\mathbb{R}}\cup \{+\infty\}$ is convex, proper, and lower semi-continuous, and $\Psi$ is continuous on any nonempty compact set $A\subseteq X$ such that $\sup\limits_{x\in A}\Psi(x) < +\infty$.

And they applied the conclusions to an elliptic variational-hemivariational inequality.

Motivatied by the above cited papers, we try to prove a multiplicity theorem of functions $f$ fulfilling the structural hypothesis $(H_f)$, mountain pass geometry and the bounded from below conditions. In Section 2, we will recall some basic definitions and preliminaries. The essential tool used in the proof is a general deformation lemma, which will be set forth in Section 3. Section 4 presents our main result, a new critical point theorem.

In the last section we consider an application to the elliptic variational-hemivariational inequality:

$(P_{\lambda}): \, \, \, \,$ Find $u\in K_{\lambda}$ such that for all $v\in K_{\lambda}$,

where $\lambda > 0$, $K_{\lambda}$ is convex and closed in $H^{1}_{0}(\Omega)$, $g: {\mathbb{R}}\rightarrow {\mathbb{R}}$ is locally bounded and measurable, and the functions $G: {\mathbb{R}}\rightarrow {\mathbb{R}}$ and $\mathcal{G}:H^{1}_{0}(\Omega)\rightarrow {\mathbb{R}}$ given by

respectively, are well defined and locally Lipschitz continuous. Some results of ^[16] are improved.

2.   Preliminaries

Let $(X, \|\cdot\|)$ be a reflexive Banach space. Denote by $B(x, \delta): = \{z\in X:\|z-x\| < \delta\}$ as well as $B_{\delta}: = B(0, \delta)$. The symbol $[x, z]$ denote the segment joining $x$ to $z$, namely $[x, z]: = \{(1-t)x+tz:t\in [0, 1]\},$ and $(x, z]: = [x, z]\backslash\{x\}$. We denote by $X^{\ast}$ the dual space of $X$, while $\langle\cdot, \cdot\rangle$ stands for the duality pairing between $X$ and $X^{\ast}$. A functional $\varphi:X\rightarrow {\mathbb{R}}\cup \{+\infty\}$ is proper if $D\varphi = \{x\in X: \varphi(x) < \infty\}\neq \emptyset$. Functional $\Phi:X\rightarrow {\mathbb{R}}$ is called locally Lipchitz continuous if for every $x\in X$ there exists a neighborhood $V_{x}$ of $x$ and a constant $L_{x}\geq0$ such that

Let $\Phi^{0}(x; z)$ be the generalized directional derivative of $\Phi$ at $x$ along the direction $z$, i.e.,

The generalized gradient of the function $\Phi$ at $x$, denoted by $\partial\Phi(x)$, is the set

The mapping $z\mapsto\Phi^{0}(x; z)$ is positively homogeneous and sub-additive, thus, due to the Hahn-Banach theorem, the set $\partial\Phi(x)$ is nonempty. In the sequel, we state the main properties of the generalized directional derivatives and the generalized gradients:

$1)$ For each $x\in X, \ \partial\Phi(x)$ is a nonempty, convex in addition to $weak^{\ast}$ compact subset of $X^*$.

$2)$ For each $x, z\in X, \; \Phi^0(x, z)$ is upper semicontinuous on $X\times X$.

$3)$ For each $x, z\in X,$ we have $\Phi^{0}(x; z) = \max\{ \langle x^{\ast}, z\rangle; \ x^{\ast}\in \partial\Phi(x)\}$.

$4)$ If $\Phi$ attains a local minimum or maximum at $x$, then $0\in \partial\Phi(x)$.

$5)$ The function $m_{\Phi}(x) = \min\{\|x^{\ast}\|_{X^{\ast}}, x^{\ast}\in \partial\Phi(x)\}$ exists and is lower semi-continuous.

Let $\Psi:X\rightarrow {\mathbb{R}}\cup \{+\infty\}$ be convex, proper, and lower semi-continuous. Set $D_{\Psi} = \{x\in X:\Phi(x) < +\infty\}$, then $\Psi$ is continuous in $int(D_{\Psi})$ (see ^[8]). To simplify notation, denote by $\partial\Psi(x)$ the subdifferential of $\Psi$ at the point $x\in X$ in the sense of convex analysis, while $D_{\partial\Psi} = \{x\in X:\partial\Psi(x)\neq\emptyset\}.$ By ^[8], $int(D_{\Psi}) = int(D_{\partial\Psi})$, $\partial\Psi(x)$ is convex and $weak^{\ast}$ closed.

Let $f$ be a function on $X$ satisfying the hypothesis$(H_f)$, $a \in {\mathbb{R}}$. Define

For every $\epsilon, r > 0$, we introduce the set

it is easy to see that $F^{r}_{a, \epsilon}$ is closed.

3.   A deformation result

In this section we establish a deformation lemma for the functions satisfying the hypothesis $(H_f)$.

Lemma 3.1. Suppose $x\in int(D_{\Psi})$. Then for every $x_{n}\rightarrow x$ in $X$ and every $z_{n}^{\ast}\in \partial\Psi(x_{n}), n\in N$, there exists $z^{\ast}\in \partial\Psi(x)$ as well as a subsequence $\{z_{r_{n}}^{\ast}\}$ of $\{z_{n}^{\ast}\}$ such that $z_{r_{n}}^{\ast}\rightharpoonup z^{\ast}$ in $X^{\ast}$.

For the proof the reader could refer to [21,Remark 2.1].

Lemma 3.2. Let $f$ be a function satisfying $(H_f)$. Assume that there exist constants $\epsilon > 0, r > 0$ and $a\in {\mathbb{R}}$ such that $F^{r}_{a, \epsilon}\neq\emptyset, \, \, F^{r}_{a, \epsilon}\subseteq int(D_{\Psi})$, and

Then for every $x\in F^{r}_{a, \epsilon}$, there exists $\xi_{x}\in X$ such that

Proof. Since $\inf\{\|x^{\ast}+z^{\ast}\|:x^{\ast}\in \partial\Phi(x), \ z^{\ast}\in \partial\Psi(x), \ x\in F^{r}_{a, \epsilon}\} > 2\epsilon$, there exists an $\epsilon_{0} > 0$ such that for every $x\in F^{r}_{a, \epsilon}, x^{\ast}\in \partial\Phi(x), z^{\ast}\in \partial\Psi(x),$ we have $\|x^{\ast}+z^{\ast}\|_{X^{\ast}}\geq2\epsilon+\epsilon_{0}$.

Fix an $x\in F^{r}_{a, \epsilon}$, since $\partial\Phi(x)$ and $\partial\Psi(x)$ are nonempty and convex, so is $\partial\Phi(x)+\partial\Psi(x)$. As $X$ is reflexive, $\partial\Phi(x)$ is weak$^*$ compact and $\partial\Psi(x)$ is weak$^*$ closed, then $\partial\Phi(x)+\partial\Psi(x)$ is closed.

Note that $0\notin\partial\Phi(x)+\partial\Psi(x)$. By [2, Corollary 3.20], we have $u^{\ast}\in \partial\Phi(x), v^{\ast}\in \partial\Psi(x)$ satisfying

Now the Hahn-Banach theorem provides a point $\xi_{x}\in X$ with $\|\xi_{x}\| = 1$ and whenever $x^{\ast}\in \partial\Phi(x), \, \, z^{\ast}\in \partial\Psi(x)$,

Since $\|u^{\ast}+v^{\ast}\|_{X^{\ast}} = \|u^{\ast}+v^{\ast}\|_{X^{\ast}}\|\xi_{x}\| = \max\Big\{\langle w^{\ast}, \xi_{x}\rangle, w^{\ast}\in \overline{B_{\delta^{\ast}}}\Big\}$, the above inequality and Lemma 3.1 lead to

The proof is completed.

Lemma 3.3 Under the conditions of Lemma 3.2, for every $x\in F^{r}_{a, \epsilon}$, there exists a $\delta_{x} > 0$ such that

where $\xi_{x}$ is given by Lemma 3.2.

Proof. If the conclusion were false, then we could find $x\in F^{r}_{a, \epsilon}, \{x_{n}^{'}\}, \{x''_{n}\}\subseteq X$ and $\{x_{n}^{\ast}\}, \{z_{n}^{\ast}\}\subseteq X^{\ast}$ such that

Due to the reflexivity of $X$ and (3.3), Proposition 2.1.2 of ^[22] yields $x^{\ast}\in X^{\ast}$ such that $x^{\ast}_{n}\rightharpoonup x^{\ast}$ in $X^{\ast}$, where a subsequence is considered when necessary, while Proposition 2.1.5 of ^[22] forces $x^{\ast}\subseteq\partial\Phi(x)$. Since $x\in F^{r}_{a, \epsilon}\subseteq int(D_{\Psi}) = int(D_{\partial\Psi})$, combining (3.4) with Lemma 3.1, we obtain, up to subsequences, $z^{\ast}_{n}\rightharpoonup z^{\ast}$ for some $z^{\ast}\in \partial\Psi(x)$. Now from (3.5) it follows, as $n\rightarrow+\infty$, $\langle x^{\ast}+z^{\ast}, \xi_{x}\rangle\leq2\epsilon$. However, this contradicts (3.1).

Theorem 3.4 Let $f$ be a function satisfying $(H_f)$, assume that there exist constants $\epsilon > 0, r > 0$ and $a\in R$ such that $F^{r}_{a, \epsilon}\neq\emptyset, F^{r}_{a, \epsilon}\subseteq int(D_{\Psi})$, and

Then there exists a continuous mapping $\eta: X\rightarrow X$ with the following properties:

Proof. The family of balls $\mathbb{B} = \{B(x, \delta_{x}):x\in F^{r}_{a, \epsilon}\}$ constructed through Lemma 3.3 represents an open covering of $F^{r}_{a, \epsilon}$, and the assumptions ensure that $F^{r}_{a, \epsilon}$ is a nonempty para-compact set because it is closed. So $\mathbb{B}$ possesses an open locally finite refinement $\mathbb{V} = \{V_{i}; i\in I\}$. Moreover, to each $i\in I$ there corresponds $\xi_{i}\in X$ such that $\|\xi_{i}\| = 1$ as well as

Shrink $\mathbb{V}$ to an open locally finite covering $\mathbb{W} = \{W_{i}; i\in I\}$ fulfilling for every $i\in I$, $W_{i}\subseteq V_{i}$ ([9,Theorems Ⅷ 2.2 and Theorems Ⅶ 6.1]) with $W_{i}$ is convex, and $f|_{W_{i}}(\cdot)$ is Lipschitz continuous satisfying

Set

We observe that $V:X\rightarrow X$ is locally Lipschitz continuous and

The existence-uniqueness theorem for ordinary differential equations provides a mapping $\sigma\in C^{0}({\mathbb{R}}\times X, X)$ such that

We claim that

In fact, if $x\in X\setminus W$, $V(x) = 0$ and thus $\sigma(\cdot, x)$ is constant and (3.9) holds true. In the case $x\in W$, we start by noting that $\sigma({\mathbb{R}}, x)\subseteq W.$ Indeed setting $T = \sup\{t > 0: \sigma((-t, t), x)\subseteq W\}$, assume by contradiction that $T < +\infty$. Hence we have $W_{x} = \sigma(T, x) = \lim_{t\rightarrow T^{-}}\sigma(t, x)\in \partial W$. Then the Cauchy problem

admits the constant solution $\widetilde{\sigma}(\cdot)\equiv W_{x}$, as does $t\mapsto \sigma(t, x)$ for $t\leq T$, which is against the uniqueness of solutions.

Fixing $t\in {\mathbb{R}}$, we know that $\sigma(t, x)\in W$ and due to the local finiteness of W, the set $J = \{i\in I: \sigma(t, x)\in W_{i}\}$ is finite. It follows that $\widetilde{W} = \cap_{i\in J}W_{i}$ is a convex, open neighborhood of $\sigma(t, x)$, and there exists $\delta > 0$ such that

For arbitrary $t', t''\in (t-\delta, t+\delta)$ with $t' < t''$. Lebourg's mean value theorem provides $y\in (\sigma(t', x), \sigma(t'', x)), \ x^{\ast}\in\partial\Phi(y), \ z^{\ast}\in\partial\Psi(y)$ satisfying

where (3.6) and (3.10) have been used. Given $p, q\in[t, t+1]$ with $p < q$, a standard compactness argument and the above estimate enable us to find $t_{1}, t_{2}, ..., t_{s}\in[t, t+1]$ with $p = t_{1} < t_{2} < ... < t_{s} = q$ such that

for all $i = 1, 2, ..., s.$ It turns out that

which establish (3.9). Now we define

From the general theory of ordinary differential equations it is well known that $\eta:X\rightarrow X$ is a homeomorphism.

Since (3.7) renders $\{x\in X:|f(x)-a|\geq 2\epsilon\}\subseteq X\setminus W$ when $x\in X\setminus W, \ V(x) = 0$, thus $\sigma(\cdot, x)$ is constant, and $\eta(x) = \sigma(1, x) = \sigma(0, x) = x$. We then deduce property (2).

From (3.8), for all $x\in X$, we have

i.e., property (3) holds true.

Since $f(\eta(x)) = f(\sigma(1, x))\leq f(\sigma(0, x)) = f(x)$, for all $x\in X$, the property (4) holds true.

For every $x\in D_\Psi$, there is $f(x) < +\infty$. Since (4) holds, $f(\eta(x))\leq f(x) < +\infty$, so $\eta(x)\in D_\Psi$. i.e. $\eta(D_\Psi)\subseteq D_\Psi$, (5) holds true.

In order to prove (6), let $x\in X$ with $\|x\|\leq r$ and $f(x)\leq a+\epsilon$. If $f(x)\leq a-\epsilon$, (6) follows from (4) immediately. In case $a-\epsilon < f(x)\le a+\epsilon$, we argue by contradiction. Suppose

In addition, through (3.8), for all $t\in[0, 1]$ we have

Consequently, $\sigma([0, 1], x)\subseteq F^{r}_{a, \epsilon}$, which forces $l(\sigma(\cdot, x))|_{[0, 1]}\equiv 1$. Then (3.11) with $p = 0$ and $q = 1$ reads as

Combining (3.12) and (3.13) gives

which is a contradiction.

4.   Existence of critical points

Fix $v_{0}, \ v_{1}\in D_{\Psi}$. Consider the following set of paths

and a function $f: X\rightarrow {\mathbb{R}}\cup\{+\infty\}$ which verifies hypothesis $(H_f)$. Set

Theorem 4.1 Suppose $f: X\rightarrow {\mathbb{R}}\cup\{+\infty\}$ satisfies $(H_f)$ and $(PS)_{f}$. Assume in addition that

$(i_{1}) \; f$ is bounded below and coercive, and put $\alpha = \inf_{x\in X}f(x)$;

$(i_{2}) \; \alpha < \max\{f(v_{0}), f(v_{1})\}\leq c, \ v_{0}\neq v_{1}$ and for every $\gamma\in \Gamma$, there exists $t\in (0, 1)$ such that $f(\gamma(t))\geq \max\{f(v_{0}), f(v_{1})\}$;

$(i_{3})$ for every $a \in {\mathbb{R}}$, there exist $r > 0$ and $\epsilon_{0} > 0$ such that $F^{r}_{a, \epsilon_{0}}\subseteq int(D_{\Psi})$.

Then the function $f$ possesses at least two critical points.

Proof. Since $f$ is coercive, for every $x\in X$ with $c-1\leq f(x)\leq c+1$, there exists a constant $k$ such that

From $(i_{3})$, for $c$ and $k > 0$, there is $\epsilon_{0} > 0$ such that

Without loss of generality, we assume $\epsilon_{0} < 1$. Let

where $0 < \epsilon < \frac{1}{2}\epsilon_{0}$ is arbitrary, and $d(t) = \min \{t, 1-t\}, \ t\in [0, 1]$.

From $(i_{2})$, we can easily verify that

Since for every $\gamma\in \Gamma$, there exists $t_0 \in (0, 1)$ such that $f(\gamma(t_0))\geq \max\{f(v_{0}), f(v_{1})\}$, one has

thus $c_{\epsilon} > \max\{f(v_{0}), f(v_{1})\}$ for every $\epsilon\in (0, \frac{1}{2}\epsilon_{0})$.

We claim that for every $\epsilon\in (0, \frac{1}{2}\epsilon_{0})$ satisfying that $\epsilon < \frac{1}{2}(c_{\epsilon}-\max\{f(v_{0}), f(v_{1})\})$ there holds

Due to the definition of $\widehat{\epsilon}$, for every $\epsilon\in (0, \widehat{\epsilon})$, we have

and $\|x\|\leq k$. It is straightforward to verify that $F^{k}_{c_{\epsilon}, \epsilon}\subseteq int(D_{\Psi}).$

To show (4.6), we argue by contradiction. If it was not true, then we would find $\epsilon\in (0, \widehat{\epsilon})$ for which Theorem 3.4 can be applied with $a = c_{\epsilon}$ and $r = k$. So there would exist a continuous mapping $\eta:X\rightarrow X$ with the properties (1)–(6) formulated in Theorem 3.4. By the definition of $\widehat{\epsilon}$, there is $\gamma_{\epsilon}\in \Gamma$ such that

which easy to verify that

and use the definition of $\widehat{\epsilon}$ again, it is straightforward to verify that $\eta(\gamma_{\epsilon}(\cdot))\in\Gamma$ for every $\epsilon$. Hence, in view of (4.5), we may consider a sequence $\{s_{\epsilon}\}$ in $[0, 1]$ such that

Since $c_{\epsilon}+\epsilon < c+2\epsilon < c+1$ and $c_{\epsilon}-\epsilon > c-\epsilon > c-1$, it implies that $\|\gamma_{\epsilon}(s_{\epsilon})\|\leq k$.

Exploiting (4.7) and property (6), we achieve the contradiction

thereby (4.6) holds true.

By virtue of (4.6), for every $x\in X$ and all $n\in N$ sufficiently large, there exists $x_{n}\in F^{k}_{c_{\frac{1}{n}}, \frac{1}{n}}$, $x^{\ast}_{n}\in \partial\Phi(x_{n}), z^{\ast}_{n}\in \partial\Psi(x_{n})$ such that

and

This guarantees that

and

Since $f$ satisfies the $(PS)_{f}$ condition, there is an $\overline{x}\in X$ such that $x_{n}\rightarrow \overline{x}$ in X, where a subsequence is considered when necessary. At this point, $\overline{x}$ is a critical point of $f$, and $\overline{x}\in K_{c}(f).$

Next we prove that $f$ possesses a global minimum point $x_{0}\in X.$ Since by $(i_1)$ and the condition $(PS)_f$, each minimizing sequence for $f$ possesses a convergent subsequence (see^[16]), the function $f$ must attain its minimum at some point $x_0 \in X$.

Due to $f(x_0) = \alpha < c = f(\bar{x})$, $x_0\neq \bar{x},$ which completes the proof.

Remark 4.2 By the above proof, one can find that under the conditions of Theorem 4.1, when $a\geq\alpha$, there exist $r > 0, \ \epsilon > 0$ such that $F^r_{a, \epsilon}\neq\emptyset.$ By the coercivity of $f$, if $\Psi$ is convex and continuous, the condition $(i_3)$ obviously holds.

5.   An application

In this section we use Theorem 4.1 to discuss an elliptic variational-hemivariational inequality in the sense of Panagiotopoulos ^[24].

Let $\Omega$ be a nonempty, bounded, open subset of ${\mathbb{R}}^{N}$ ($N\geq3$) with smooth boundary $\partial\Omega$. Denote by $H^{1}_{0}(\Omega)$ the usual Sobolev space with norm

It's well known that for $p\in [1, 2^{\ast}]$, $2^{\ast} = 2N/(N-2)$, there exists a positive constant $C_{p}$ such that

Given a function $a\in L^{\infty}(\Omega)$ satisfying $a(x)\geq0$ for a.e. $x\in \Omega$. Let

If $g: {\mathbb{R}}\rightarrow {\mathbb{R}}$ satisfies the condition

$(g_{1}) \ \ g$ is locally bounded and measurable.

Then the functions $G: {\mathbb{R}}\rightarrow {\mathbb{R}}$ and $\mathcal{G}:H^{1}_{0}(\Omega)\rightarrow {\mathbb{R}}$ given by

respectively, are well defined and locally Lipschitz continuous. So it makes sense to consider their generalized directional derivatives $G^{0}$ and $\mathcal{G}^{0}$. On account of [22,formula(9),P.84] one has

We will further assume

$(g_{2}) \ \ \lim\limits_{t\rightarrow 0}\frac{g(t)}{t} = 0$;

$(g_{3}) \ \ \limsup\limits_{|t|\rightarrow+\infty}\frac{g(t)}{t}\leq 0$;

$(g_{4})$ there exists a $\xi_{0}\in {\mathbb{R}}$ such that $G(\xi_{0}) < 0.$

Through $(g_{3})$ for every $\epsilon > 0$ there exists a constant $r > 0$ such that

Since $g$ is locally bounded, we also have

Let $\lambda > 0$, $\mu(\Omega)$ be the Lebesgue measure of $\Omega$. Define

A set $K_{\lambda}\subseteq H^{1}_{0}(\Omega)$ is called of type $(K^{g}_{\lambda})$ provided

$(K^{g}_{\lambda}): K_{\lambda}$ is convex and closed in $H^{1}_{0}(\Omega)$. Moreover, $\overline{B_{r_{\lambda}}}\subseteq K_{\lambda}.$

Given $\lambda > 0$ and $K_{\lambda}$ satisfying $(K^{g}_{\lambda})$, consider the elliptic variational-hemivariational inequality problems:

$(P_{\lambda}):$ Find $u\in K_{\lambda}$ such that for all $v\in K_{\lambda}$,

Due to (5.2), any solution $u$ of $(P_{\lambda})$ also fulfills the inequality

If g is continuous and $K_{\lambda} = H^{1}_{0}(\Omega)$, the function $u\in H^{1}_{0}(\Omega)$ turns out a weak solution to the Dirichlet problem

which has been previously investigated in ^[3,14] under more restrictive conditions.

Theorem 5.1 Suppose $(g_{1})-(g_{4})$ hold true. Then, for every $\lambda$ sufficiently large, problem $(P_{\lambda})$ possesses at least two solutions.

Proof. Let $X = H^{1}_{0}(\Omega)$, $p \in (2, 2^{\ast})$. Define a functional $f(u) = \Phi(u)+\Psi(u)$ on $X$ as follows:

as well as

where $\lambda > 0$ and $K_{\lambda}\subseteq H^{1}_{0}(\Omega)$ is of type $(K^{g}_{\lambda})$. Owing to $(g_{1})$ the function $\Phi:X\rightarrow {\mathbb{R}}$ turns out locally Lipschitz continuous. Consequently, $f$ satisfies condition $(H_f)$.

We shall prove that $f$ is bounded from below and coercive.

By (5.3) and (5.4), one has

Which clearly implies

Then we obtain

Setting $\epsilon\in(0, \frac{\beta}{\lambda})$, then we have

which shows the claim.

Let us next show that the function $f$ satisfies condition $(PS)_{f}$. Pick a sequence $\{u_{n}\}\subseteq X$ such that $\{f(u_{n})\}$ is bounded and

for all $\ n\in N, v\in X$, where $\epsilon_{n}\rightarrow 0^{+}$.

By (5.7) one evidently has $\{u_{n}\}\subseteq K_{\lambda}$, and $\{f(u_{n})\}$ is bounded. Since $f$ is coercive, the sequence $\{u_{n}\}$ turns out bounded. Passing to a subsequence if necessary, we suppose $u_{n}\rightharpoonup u$ in X and $u_{n}\rightarrow u$ in $L^{2}(\Omega)$. The point $u$ belongs to $K_{\lambda}$ because this set is weakly closed.

Exploiting (5.7) with $v = u$, we then get

for all $\ n\in N.$

From $u_{n}\rightharpoonup u$ in $X$ it follows

The upper semi-continuity of $\mathcal{G}^{0}$ on $L^{2}(\Omega)\times L^{2}(\Omega)$ forces

Taking account of (5.9), (5.10) besides $\{\|u-u_n\|\}$ is bounded, and letting $n\rightarrow+\infty$, inequality (5.8) yields

Hence, thanks to [2,Proposition Ⅲ.3], $u_{n}\rightarrow u$ in $X$. i.e., $(PS)_{a}$ holds.

By $(g_{4})$, we can construct an $u_{0}\in X$ such that $\mathcal{G}(u_{0}) < 0$. Moreover, $u_{0}\in \overline{B_{r_{\lambda}}}$ for any $\lambda\geq \frac{1}{4}\|u_{0}\|^{2}$. Therefore, $\inf_{u\in X}f(u)\leq f(u_{0}) < 0$ provided

while $f(0) = \lambda\mathcal{G}(0) = 0.$

Our next objective is to verify $(i_{1})$. From $(g_{2})$, there exists $\sigma\in (0, r)$ such that

Due to (5.5), one has

provided $|\xi|\geq\sigma$.

The Sobolev embedding theorem gives

where $C^{\ast} = (\frac{Mr}{\sigma^{p}}+\frac{\epsilon}{2\sigma^{p-2}})C^{p}_{p}$. Then by (5.11) (5.12) and (5.1) we get

Let us next prove that for a suitable constant $\theta > 0$,

Indeed, if it's not true, there exists a sequence $\{u_{n}\}\subseteq X$ enjoying the properties

Passing to a subsequence if necessary, we may suppose $u_{n}\rightharpoonup u$ in $X$ as well as $u_{n}\rightarrow u$ in $L^{2}(\Omega)$. Thus, letting $n\rightarrow +\infty$ in $(5.15)$ yields

Using the sobolev embedding theorem and $\beta = ess\inf_{x\in \Omega}a(x)\geq0$ we obtain

Gathering (5.16) and (5.17) together, leads to $u = 0$. By (5.15) this forces $u_{n}\rightarrow 0$ in $X$, against to $\|u_{n}\| = 1, \forall n\in N$.

Combining (5.14) with (5.13), provides

Pick $\epsilon > 0$ and $R\in (0, \frac{1}{2}\|u_{0}\|)$ sufficiently small such that

Then by (5.18) we have

Furthermore, it is easy to prove that $R < \frac{1}{2}\|u_{0}\| < r_{\lambda}$.

Now, let $v_{0} = 0, \ v_{1} = u_{0}$. Define

Thanks to (5.19) and the definition of $c$, one has

and for every $\gamma\in \Gamma$, there exists a $t\in (0, 1)$ such that $\gamma(t)\in X$ and $\|\gamma(t)\| = R.$ Then by (5.19) again, we obtain $f(\gamma(t))\geq 0.$ Hence hypothesis $(i_{1})$ of Theorem 4.1 is fulfilled.

Finally, let us prove that $(i_{3})$ holds. Since $f$ is bounded below, put $\alpha = \inf_{x\in X}f(x)$, then $\alpha < 0\leq c$. For every $a\geq \alpha$ suppose that $a < \lambda$, then there exist $r > 0$ and $\epsilon_{0} > 0$ such that

Indeed, there is $\epsilon_{0} > 0$ such that $a+\epsilon_{0}\leq\lambda < 2\lambda$.

Inequality (5.6) ensures that

So we immediately have $\{u\in X: f(u)\leq a+\epsilon_{0}\}\subseteq int (D_{\Psi}).$

Since $f$ is coercive, there exists $r > 0$ such that every $u\in X$ satisfies $a-\epsilon_{0}\leq f(u)\leq a+\epsilon_{0}$, and $\|u\|\leq r+1$, which leads to $(5.20)$, i.e., condition $(i_3)$ holds true.

We are now in a position to apply Theorem 4.1. By this theorem, there exist at least two points $u_{1}, u_{2}\in X$ such that

The choice of $\Psi$ gives both $u_{i}\in K_{\lambda}$ and $\Phi^{0}(u_{i}; v-u_{i})\geq 0, \ v\in K_{\lambda}, \ i = 1, 2.$ Namely, $u_{1}, u_{2}$ are solutions to the problem $(P_{\lambda})$.

Example 5.2 The aim of this example is to exhibit a nontrivial case of set in $H^{1}_{0}(\Omega)$ of type $(K^{g}_{\lambda}).$ Let $h: H^{1}_{0}(\Omega)\rightarrow {\mathbb{R}}$ be a weakly continuous and convex function. For $\bar{r} > 0$ fixed, $\lambda > 0,$ put

with the same notation as before. The ball $\bar{B}(0, \bar{r}_{\lambda})$ is a weakly compact subset of $H^{1}_{0}(\Omega)$, since $h$ is weakly continuous, there exists $u_{0}\in \bar{B}(0, \bar{r}_{\lambda})$ such that

i.e., $h_{\bar{B}(0, \bar{r}_{\lambda})}$ admits a global maximum. Then the set

is a subset of $H^{1}_{0}(\Omega)$ of type $(K^{g}_{\lambda}).$

Example 5.3 There exist functionals satisfying the conditions of Theorem 5.1. For example

Acknowledgments

The authors are grateful to the anonymous referees for their careful reading and helpful comments, which greatly improve the manuscript. The work was supported by the NSF of Shandong Province (No. ZR2018PA006, ZR2017PA001) and the NSF of China (No. 11901240).

Conflict of interest

The authors declare no conflict of interest.